Consider two cases as below.

Case 1: A company buys 1000 pieces per year of a certain part from vendor 'X'. The changeover time is 2 hours and the price is Rs. 10 per piece. The holding cost rate per part is 10% per year.

Case 2: For the same part, another vendor 'Y" offers a design where the changeover time is 6 minutes, with a price of Rs. 5 per piece, and a holding cost rate per part of 100% per year. The order size is 800 pieces per year from 'X' and 200 pieces per year from 'Y’.

Assume the cost of downtime as Rs. 200 per hour. The percentage reduction in the annual cost for Case 2, as compared to Case 1 is___________ (round off to 2 decimal places).

The correct answer is 5.32.

To find the percentage reduction in the annual cost for Case 2 compared to Case 1, we need to calculate the total annual cost for both scenarios. The total annual cost is composed of three components:
1. Purchase Cost: Calculated as the order size multiplied by the price per piece.
2. Holding Cost: Calculated as the average inventory level multiplied by the holding cost rate per part.
3. Downtime (Changeover) Cost: Calculated as the cost of downtime per hour multiplied by the changeover time in hours.

Let's define the general formula for the total cost:
$$T_c=(Q\times C)+\left(\frac{Q}{2}\times C\times r\right)+(\text{Downtime Cost/hour}\times \text{Changeover Time})$$
where:
- $Q$ is the annual order size (pieces per year).
- $C$ is the cost price per piece.
- $r$ is the holding cost rate per year.

Step 1: Calculate the Total Annual Cost for Case 1 ($T_{c1}$)
For Case 1, the company buys 1000 pieces per year from vendor X. The parameters are:
- $Q=1000$ pieces per year
- $C=\text{Rs.}10$ per piece
- $r=10\%=0.10$
- Changeover time = 2 hours
- Downtime cost = Rs. 200 per hour
Substituting these values into our cost formula:
$$T_{c1}=1000\times 10+\frac{1000}{2}\times 10\times 0.10+200\times 2$$
$$T_{c1}=10000+500+400=\text{Rs.}10900$$

Step 2: Calculate the Total Annual Cost for Case 2 ($T_{c2}$)
In Case 2, the order size is split between vendor X (800 pieces per year) and vendor Y (200 pieces per year). We calculate the individual costs for each vendor and then sum them up.

Cost from Vendor X ($T_{cx}$)
Only the quantity changes to 800; other vendor X specific parameters remain the same:
- $Q=800$ pieces per year
- $C=\text{Rs.}10$ per piece
- $r=10\%=0.10$
- Changeover time = 2 hours
$$T_{cx}=800\times 10+\frac{800}{2}\times 10\times 0.10+200\times 2$$
$$T_{cx}=8000+400+400=\text{Rs.}8800$$

Cost from Vendor Y ($T_{cy}$)
The parameters for vendor Y are:
- $Q=200$ pieces per year
- $C=\text{Rs.}5$ per piece
- $r=100\%=1.0$
- Changeover time = 6 minutes = $\frac{6}{60}=0.1$ hour
$$T_{cy}=200\times 5+\frac{200}{2}\times 5\times 1.0+200\times 0.1$$
$$T_{cy}=1000+500+20=\text{Rs.}1520$$

Total Cost for Case 2 ($T_{c2}$)
Summing the costs of X and Y:
$$T_{c2}=T_{cx}+T_{cy}=8800+1520=\text{Rs.}10320$$

Step 3: Calculate the Percentage Reduction
The percentage reduction in the annual cost from Case 1 to Case 2 is calculated as:
$$\text{\% Reduction}=\frac{T_{c1}-T_{c2}}{T_{c1}}\times 100$$
Substituting the total costs:
$$\text{\% Reduction}=\frac{10900-10320}{10900}\times 100$$
$$\text{\% Reduction}=\frac{580}{10900}\times 100\approx 5.3211\%$$
Rounding off to two decimal places gives a percentage reduction of 5.32%.